Under whose rule did the Mughal empire achieve its largest territorial limits?

The correct answer is Aurangzeb.

The Mughal Empire reached its greatest territorial extent during the reign of the sixth Mughal Emperor, Aurangzeb (Muhi-ud-Din Muhammad), who ruled from 1658 to 1707.
During his long reign of nearly 49 years, Aurangzeb conducted extensive military campaigns, particularly in Southern India (the Deccan region).

Key details of the territorial expansion include:
1. Conquest of the Deccan: Aurangzeb annexed the rich and powerful Deccan sultanates of Bijapur in 1686 and Golconda in 1687.
2. Northeastern Expansion: Under his general Mir Jumla, the Mughals also pushed into Assam and captured the Ahom capital, Garhgaon.
3. Subcontinental Reign: At its peak under Aurangzeb, the Mughal Empire stretched from Kabul (present-day Afghanistan) in the west to Assam in the east, and from Kashmir in the north to Jinji (in modern-day Tamil Nadu) in the south, ruling over a population of over 150 million subjects.